The Empirical Rule

68-95-99.7 Rule for Normal Distributions

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The Empirical Rule is a quick way to estimate how data are spread out when the distribution is approximately normal, or bell shaped. It tells you what percent of values lie within $1$ , $2$ , and $3$ standard deviations of the mean. This matters because many real measurements, test scores, and natural variations are modeled well by a normal distribution. With one mean and one standard deviation, you can make fast predictions about typical and unusual values.

In a normal distribution, the mean sits at the center of the bell curve and the curve is symmetric on both sides. The Empirical Rule says about $68\%$ of values fall within $1$ standard deviation of the mean, about $95\%$ within $2$ , and about $99.7\%$ within $3$ . These percentages help you estimate probabilities, identify outliers, and interpret data without calculating every area exactly. The rule is most useful when the histogram of the data looks roughly mound shaped and symmetric.

Key Facts

About 68% of data in a normal distribution lie between μ - σ and μ + σ.
About 95% of data in a normal distribution lie between μ - 2σ and μ + 2σ.
About 99.7% of data in a normal distribution lie between μ - 3σ and μ + 3σ.
Standard score formula: z = (x - μ)/σ
For a normal distribution, $\text{mean} = \text{median} = \text{mode}$ .
About 5% of data lie outside μ ± 2σ, and about 0.3% lie outside μ ± 3σ.

Vocabulary

Normal distribution: A symmetric, bell-shaped distribution where values cluster around the center and become less common farther away.
Mean: The average value of a data set, which is the center of a normal distribution.
Standard deviation: A measure of how spread out the data are from the mean.
Empirical Rule: A rule stating that about 68%, 95%, and 99.7% of normal data fall within $1$ , $2$ , and $3$ standard deviations of the mean.
z-score: The number of standard deviations a value is above or below the $\text{mean}$ .

Common Mistakes to Avoid

Using the Empirical Rule for any data set, even when the distribution is skewed or irregular. This is wrong because the rule only works well for data that are approximately normal.
Confusing the $\text{mean}$ with the $\text{standard deviation}$ , then placing interval boundaries at the wrong values. This is wrong because the $\text{mean}$ gives the center, while the $\text{standard deviation}$ gives the distance from the center.
Thinking 95% means exactly 95 out of every 100 values must fall within $2$ standard deviations. This is wrong because the rule gives an approximate percentage for a normal model, not a guaranteed count in every sample.
Forgetting that the intervals are symmetric around the $\text{mean}$ . This is wrong because in a normal distribution you must go the same number of standard deviations to the left and right of $\mu$ .

Practice Questions

1 A test score distribution is approximately normal with $\text{mean}$ $70$ and $\text{standard deviation}$ $8$ . According to the Empirical Rule, about what percent of students scored between $62$ and $78$ ?
2 The heights of a plant species are approximately normal with $\text{mean}$ $45$ cm and $\text{standard deviation}$ $3$ cm. Give the interval that contains about $95\%$ of the plants.
3 A data set is strongly right-skewed. Explain whether the Empirical Rule should be used and why.